Update: I found a way to stay logged in now although it's still not the best. Sign out of your dashboard. This will take you back to the sign in page. Sign back in, but this time make sure NOT to check that box that says "Stay signed in."
I don't know why, but many people have reported that if you uncheck that box, you'll be able to stay logged in so that you can commment on other people's blogs. Of course, whenever you shut down your browser it will meaning signing in all over again next time you want to comment on someone's blog.
Also, people who have this problem also will not be able to see their followers anymore. I can see that I have followers, but I can't see who they are.
If this is happening to you, there's a thread about it at Google's help forum here: Blogger Problems
Edited to say that I also can no longer see Facebook "Notes" on other people's Facebook accounts either if they have a lot of them. I'm guessing this is all due to a problem with the latest Java update.